Maslov Index on Symplectic Manifolds. With Supplement by A. T. Fomenko “Constructing the Generalized Maslov Class for the Total Space $W=\mathbb{T}^*(M)$ of the Cotangent Bundle”

Abstract

The geometric properties of the Maslov index on symplectic manifolds are discussed. The Maslov index is constructed as a homological invariant on a Lagrangian submanifold of a symplectic manifold. In the simplest case, a Lagrangian submanifold $\Lambda\subset \mathbb{R}^{2n}\approx \mathbb{R}^{n}\oplus\mathbb{R}^{n}$ is a submanifold in the symplectic space $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$, in which the symplectic structure is given by the nondegenerate form $\omega=\sum_{i=1}^n dx^{i}\wedge dy^{i}$ and $\Lambda\subset\mathbb{R}^{2n}$ is a submanifold, $\dim\Lambda=n$, on which the form $\omega$ is trivial. In the general case, a symplectic manifold $(W,\omega)$ and the bundle of Lagrangian Grassmannians $\mathcal{LG}(\mathbb{T}W)$ is considered. The question under study is as follows: when is the Maslov index, given on an individual Lagrangian manifold as a one-dimensional cohomology class, the image of a one-dimensional cohomology class of the total space $\mathcal{LG}(\mathbb{T}W)$ of bundles of Lagrangian Grassmannians? An answer is given for various classes of bundles of Lagrangian Grassmannians.